Subgroup ($H$) information
| Description: | not computed |
| Order: | \(531441\)\(\medspace = 3^{12} \) |
| Index: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| Exponent: | not computed |
| Generators: |
$\langle(7,8,9)(10,11,12)(13,14,15)(28,30,29)(31,33,32), (28,30,29)(31,32,33), (13,15,14) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.D_4:D_4$ |
| Order: | \(705277476864\)\(\medspace = 2^{14} \cdot 3^{16} \) |
| Exponent: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $A_4^2\wr C_2.C_2^2.D_4$ |
| Order: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Automorphism Group: | $C_2^8.S_3\wr D_4$, of order \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(2821109907456\)\(\medspace = 2^{16} \cdot 3^{16} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |